= Nonsurjective self-map of real projective space is null-homotopic
For $n\ge2$, a nonsurjective self-map of <Real projective space> $\mathbb{RP}^n$ has zero <mod-two degree of a map between closed manifolds>. In the <mod-two cohomology ring of real projective space>, the nonzero degree-one class $a$ has $a^n\ne0$. If pullback fixed $a$, it would fix $a^n$, contradicting zero degree. Thus the map is trivial on the <fundamental group> and lifts to $S^n$. A point missed by the original map has both its inverse images missed by the lift. The lift therefore takes values in a punctured sphere, which is contractible, and is null-homotopic. Composing its contraction with the covering projection proves the conclusion.
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